OFFSET

1,3

COMMENTS

The transformation process starts with an upwards column of numbers 1..n.

The rightmost number of the topmost row slides right across columns 1 smaller (if any), and then drops down either onto a 2 or more smaller final column if there is one, or otherwise starting a new final column.

Sliding steps continue until reaching a single row (all columns height 1), which is triangle row n.

The final move is always with 2 that slides rightward then one step down to the end of the resulting single row. 1 never moves.

The resulting row is a self-inverse permutation (involution) because the reverse steps are exactly those performed if the row is stood up as the starting column again.

In the case of transforming the column of a composite n, there will be instances when a number sliding sideways will drop down just one step as the changing stack becomes a complete rectangle of rows and columns. Such a number is always greater by 1 than the height of the completed rectangle, whose height and width are divisors of n.

The travel of a sliding number, other than 2, that thus completes a rectangle will continue: it immediately moves sideways again one step, then downwards. Those n's that are prime numbers will not have such number since their changing stack can never form a rectangle with divisors greater than 1, except themselves.

When performing the process in an orthogonal grid where the numbers slide sideways and downward in discrete steps, the total steps for n is an oblong number, the sequence of which is A002378.

In the orthogonal grid, in mid-process, the bounding box of the changing stack of numbers follows the n = x*y curve, and is in exact contact with it at integer x and y points where n is composite.

LINKS

Thomas Scheuerle, Numbers colored by absolute displacement. Horizontally: row n of the triangle T(n, k). Vertically: k.

Thomas Scheuerle, Numbers colored by value. Horizontally: row n of the triangle T(n, k). Vertically: k.

FORMULA

a(floor(((n+3)^2 - 2*n - 3)/2)) = 3, for n > 0. - Thomas Scheuerle, Mar 21 2023

EXAMPLE

Triangle T(n,k) begins:

n/k | 1 2 3 4 5 6 7

----------------------------

1 | 1;

2 | 1, 2;

3 | 1, 3, 2;

4 | 1, 4, 3, 2;

5 | 1, 5, 3, 4, 2;

6 | 1, 6, 4, 3, 5, 2;

7 | 1, 7, 4, 3, 5, 6, 2;

...

.

A few snapshots of the process for n = 7, a prime number:

.

7

6

5

4 4

3 3 5 3 5 3

2 2 6 2 6 2 6 5 2 6 5

1 1 7 1 7 4 1 7 4 1 7 4 3 1 7 4 3 5 6 2

.

An example showing some stages of the process for a composite n = 6, with completed rectangles:

.

6

5

4

3 3 4

2 2 5 2 5 3

1 1 6 1 6 4 1 6 4 3 5 2

.

Step-by-step animation frames, showing 8, the rightmost number of the top row, sliding and dropping during its second movement, in the operation for n = 11:

.

4 8 4 8 4 8 4 4 4

3 9 5 3 9 5 3 9 5 3 9 5 8 3 9 5 3 9 5

2 10 7 2 10 7 2 10 7 2 10 7 2 10 7 8 2 10 7

1 11 6 1 11 6 1 11 6 1 11 6 1 11 6 1 11 6 8

PROG

(MATLAB)

function a = A361642( max_row )

a = 1;

for r = 2:max_row

p = [1:r];

for k = 2:r-1

j = [1:r];

t1 = find(mod(j, k) == 0);

t2 = find(mod(j, k) ~= 0);

j(t1) = [r:-1:r-length(t1)+1];

j(t2) = [1:length(t2)];

p = p(j);

end

a = [a p];

end

end % Thomas Scheuerle, Mar 21 2023

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Tamas Sandor Nagy, Mar 19 2023

STATUS

approved